Fit log returns to F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated shape parameter (degrees of freedom).
xi is the estimated skewness parameter.

Log returns data 2011-2023.

For 2011, medium risk data is used in the high risk data set, as no high risk fund data is available prior to 2012.
vmrl is a long version of Velliv medium risk data, from 2007 to 2023. For 2007 to 2011 (both included) no high risk data is available.

Summary of gross returns

##       vmr             pmr             mmr             vhr       
##  Min.   :0.868   Min.   :0.904   Min.   :0.988   Min.   :0.849  
##  1st Qu.:1.044   1st Qu.:1.042   1st Qu.:1.013   1st Qu.:1.039  
##  Median :1.097   Median :1.084   Median :1.085   Median :1.099  
##  Mean   :1.070   Mean   :1.065   Mean   :1.066   Mean   :1.085  
##  3rd Qu.:1.136   3rd Qu.:1.107   3rd Qu.:1.101   3rd Qu.:1.160  
##  Max.   :1.168   Max.   :1.141   Max.   :1.133   Max.   :1.214  
##       phr             mhr       
##  Min.   :0.878   Min.   :0.977  
##  1st Qu.:1.068   1st Qu.:1.013  
##  Median :1.128   Median :1.113  
##  Mean   :1.095   Mean   :1.087  
##  3rd Qu.:1.182   3rd Qu.:1.128  
##  Max.   :1.208   Max.   :1.207
##       vmrl      
##  Min.   :0.801  
##  1st Qu.:1.013  
##  Median :1.085  
##  Mean   :1.061  
##  3rd Qu.:1.128  
##  Max.   :1.193
##            vmr   pmr   mmr   vhr   phr   mhr
## Min.   : 0.868 0.904 0.988 0.849 0.878 0.977
## 1st Qu.: 1.044 1.042 1.013 1.039 1.068 1.013
## Median : 1.097 1.084 1.085 1.099 1.128 1.113
## Mean   : 1.070 1.065 1.066 1.085 1.095 1.087
## 3rd Qu.: 1.136 1.107 1.101 1.160 1.182 1.128
## Max.   : 1.168 1.141 1.133 1.214 1.208 1.207
Min. : ranking 1st Qu.: ranking Median : ranking Mean : ranking 3rd Qu.: ranking Max. : ranking
0.988 mmr 1.068 phr 1.128 phr 1.095 phr 1.136 vmr 1.168 vmr
0.977 mhr 1.044 vmr 1.113 mhr 1.087 mhr 1.107 pmr 1.141 pmr
0.904 pmr 1.042 pmr 1.099 vhr 1.085 vhr 1.101 mmr 1.133 mmr
0.878 phr 1.039 vhr 1.097 vmr 1.070 vmr 1.160 vhr 1.214 vhr
0.868 vmr 1.013 mmr 1.085 mmr 1.066 mmr 1.182 phr 1.208 phr
0.849 vhr 1.013 mhr 1.084 pmr 1.065 pmr 1.128 mhr 1.207 mhr 
## cov(vmr, pmr) =  -0.001094875
## cov(vhr, phr) =  -0.0001730651

Velliv medium risk, 2011 - 2023

## 
## AIC: -27.8497 
## BIC: -25.58991 
## m: 0.0480931 
## s: 0.1198426 
## nu (df): 3.303595 
## xi: 0.03361192 
## R^2: 0.993 
## 
## An R^2 of 0.993 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 7.4 percent
## What is the risk of losing max 25 %? =< 1.8 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 41 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Log returns for Velliv medium risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 278.437 kr.
## SD of portfolio index value after 20 years: 122.559 kr.
## Min total portfolio index value after 20 years: 1.432 kr.
## Max total portfolio index value after 20 years: 960.495 kr.
## 
## Share of paths finishing below 100: 5.16 percent

Velliv medium risk, 2007 - 2023

Fit to skew t distribution

## 
## AIC: -34.35752 
## BIC: -31.02467 
## m: 0.05171176 
## s: 0.1149408 
## nu (df): 2.706099 
## xi: 0.5049945 
## R^2: 0.978 
## 
## An R^2 of 0.978 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 5.4 percent
## What is the risk of losing max 25 %? =< 1.3 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 36.2 percent
## What is the chance of gaining min 25 %? >= 0.3 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks good. Log returns for Velliv high risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened. But because the disastrous loss in 2008 was followed by a large profit the following year, we see some increased upside for the top percentiles. Beware: A 1.2 return following a 0.8 return doesn’t take us back where we were before the loss. Path dependency! So if returns more or less average out, but high returns have a tendency to follow high losses, that’s bad!

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
## 
## Mean portfolio index value after 20 years: 292.991 kr.
## SD of portfolio index value after 20 years: 116.159 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 1003.923 kr.
## 
## Share of paths finishing below 100: 3.18 percent

Velliv high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -21.42488 
## BIC: -19.16508 
## m: 0.06471454 
## s: 0.1499924 
## nu (df): 3.144355 
## xi: 0.002367034 
## R^2: 0.991 
## 
## An R^2 of 0.991 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 8.3 percent
## What is the risk of losing max 25 %? =< 2.5 percent
## What is the risk of losing max 50 %? =< 0.4 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 53.3 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Returns for Velliv medium risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
## 
## Mean portfolio index value after 20 years: 405.817 kr.
## SD of portfolio index value after 20 years: 217.397 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 1483.003 kr.
## 
## Share of paths finishing below 100: 3.86 percent

PFA medium risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -33.22998 
## BIC: -30.97018 
## m: 0.05789224 
## s: 0.1234592 
## nu (df): 2.265273 
## xi: 0.477324 
## R^2: 0.991 
## 
## An R^2 of 0.991 suggests that the fit is extremely good.
## 
## What is the risk of losing max 10 %? =< 3.3 percent
## What is the risk of losing max 25 %? =< 0.9 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 32.7 percent
## What is the chance of gaining min 25 %? >= 0.1 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks great. Log returns for PFA medium risk seems to be consistent with a skewed t-distribution.

##  [1] -0.091256521 -0.003731241  0.027312079  0.045808232  0.059068633
##  [6]  0.069575113  0.078454727  0.086316936  0.093536451  0.100370932
## [11]  0.107018607  0.114081432  0.127604387

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous. While there is some uptick at the top percentiles, the curve basically flattens out.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
## 
## Mean portfolio index value after 20 years: 319.768 kr.
## SD of portfolio index value after 20 years: 104.054 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 999.664 kr.
## 
## Share of paths finishing below 100: 2.07 percent

PFA high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -23.72565 
## BIC: -21.46585 
## m: 0.08386034 
## s: 0.1210107 
## nu (df): 3.184569 
## xi: 0.01790306 
## R^2: 0.964 
## 
## An R^2 of 0.964 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 5.3 percent
## What is the risk of losing max 25 %? =< 1.4 percent
## What is the risk of losing max 50 %? =< 0.2 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 59.6 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks ok. Returns for PFA high risk seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 555.977 kr.
## SD of portfolio index value after 20 years: 244.76 kr.
## Min total portfolio index value after 20 years: 0.626 kr.
## Max total portfolio index value after 20 years: 1748.121 kr.
## 
## Share of paths finishing below 100: 0.79 percent

Mix medium risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -36.9603 
## BIC: -34.7005 
## m: 0.05902873 
## s: 0.08757749 
## nu (df): 2.772621 
## xi: 0.02904471 
## R^2: 0.89 
## 
## An R^2 of 0.89 suggests that the fit is not completely random.
## 
## What is the risk of losing max 10 %? =< 3.3 percent
## What is the risk of losing max 25 %? =< 0.7 percent
## What is the risk of losing max 50 %? =< 0.1 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 35.6 percent
## What is the chance of gaining min 25 %? >= 0 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The fit suggests big losses for the lowest percentiles, which are not present in the data.
So the fit is actually a very cautious estimate.

Data vs fit

Let’s plot the fit and the observed returns together.

Interestingly, the fit predicts a much bigger “biggest loss” than the actual data. This is the main reason that R^2 is 0.90 and not higher.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that for a few observations out of a 1000, the losses are disastrous, while the upside is very dampened.

Monte Carlo

Version a: Simulation from estimated distribution of returns of mix.

## Down-and-out simulation:
## Probability of down-and-out: 0.01 percent
## 
## Mean portfolio index value after 20 years: 324.319 kr.
## SD of portfolio index value after 20 years: 97.388 kr.
## Min total portfolio index value after 20 years: 0 kr.
## Max total portfolio index value after 20 years: 673.26 kr.
## 
## Share of paths finishing below 100: 0.97 percent

Version b: Mix of simulations from estimated distribution of returns from individual funds.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 301.885 kr.
## SD of portfolio index value after 20 years: 82.893 kr.
## Min total portfolio index value after 20 years: 18.971 kr.
## Max total portfolio index value after 20 years: 1997.325 kr.
## 
## Share of paths finishing below 100: 0.33 percent

Mix high risk, 2011 - 2023

Fit to skew t distribution

## 
## AIC: -24.26084 
## BIC: -22.00104 
## m: 0.0822419 
## s: 0.07129843 
## nu (df): 89.86289 
## xi: 0.7697502 
## R^2: 0.961 
## 
## An R^2 of 0.961 suggests that the fit is very good.
## 
## What is the risk of losing max 10 %? =< 0.9 percent
## What is the risk of losing max 25 %? =< 0 percent
## What is the risk of losing max 50 %? =< 0 percent
## What is the risk of losing max 90 %? =< 0 percent
## What is the risk of losing max 99 %? =< 0 percent
## 
## What is the chance of gaining min 10 %? >= 46.1 percent
## What is the chance of gaining min 25 %? >= 1.2 percent
## What is the chance of gaining min 50 %? >= 0 percent
## What is the chance of gaining min 90 %? >= 0 percent
## What is the chance of gaining min 99 %? >= 0 percent

QQ Plot

The qq plot looks good Returns for mixed medium risk portfolios seems to be consistent with a skewed t-distribution.

Data vs fit

Let’s plot the fit and the observed returns together.

Estimated distribution

Now lets look at the CDF of the estimated distribution for each 0.1% increment between 0.5% and 99.5% for the estimated distribution:

We see that the high risk mix provides a much better upside and smaller downside.

Monte Carlo

Version a: Simulation from estimated distribution of returns of mix.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 501.245 kr.
## SD of portfolio index value after 20 years: 158.271 kr.
## Min total portfolio index value after 20 years: 129.151 kr.
## Max total portfolio index value after 20 years: 1335.386 kr.
## 
## Share of paths finishing below 100: 0 percent

Version b: Mix of simulations from estimated distribution of returns from individual funds.

## Down-and-out simulation:
## Probability of down-and-out: 0 percent
## 
## Mean portfolio index value after 20 years: 475.563 kr.
## SD of portfolio index value after 20 years: 163.327 kr.
## Min total portfolio index value after 20 years: 25.692 kr.
## Max total portfolio index value after 20 years: 1204.922 kr.
## 
## Share of paths finishing below 100: 0.15 percent

Compare pension plans

Risk of max loss of x percent for a single period (year).
x values are row names.

Velliv_medium Velliv_medium_long Velliv_high PFA_medium PFA_high mix_medium mix_high
0 21.3 18.2 19.9 12.2 14.3 12.7 13.0
5 12.5 9.6 12.8 6.0 8.6 6.2 4.2
10 7.4 5.4 8.3 3.3 5.3 3.3 0.9
25 1.8 1.3 2.5 0.9 1.4 0.7 0.0
50 0.2 0.2 0.4 0.2 0.2 0.1 0.0
90 0.0 0.0 0.0 0.0 0.0 0.0 0.0
99 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Chance of min gains of x percent for a single period (year).
x values are row names.

Velliv_medium Velliv_medium_long Velliv_high PFA_medium PFA_high mix_medium mix_high
0 78.7 81.8 80.1 87.8 85.7 87.3 87.0
5 63.8 64.9 69.2 71.5 75.8 71.4 69.9
10 41.0 36.2 53.3 32.7 59.6 35.6 46.1
25 0.0 0.3 0.0 0.1 0.0 0.0 1.2
50 0.0 0.0 0.0 0.0 0.0 0.0 0.0
100 0.0 0.0 0.0 0.0 0.0 0.0 0.0

MC risk percentiles: Risk of loss from first to last period.
_a is simulation from estimated distribution of returns of mix.
_b is mix of simulations from estimated distribution of returns from individual funds.

_m is medium.
_h is high.

Velliv_m Velliv_m_long Velliv_h PFA_m PFA_h mix_m_a mix_h_a mix_m_b mix_h_b
0 5.16 3.18 3.86 2.07 0.79 0.97 0 0.33 0.15
5 4.38 2.79 3.50 1.91 0.65 0.87 0 0.24 0.13
10 3.87 2.44 3.20 1.65 0.61 0.73 0 0.18 0.12
25 2.62 1.60 2.15 1.30 0.42 0.43 0 0.09 0.07
50 1.11 0.72 0.99 0.62 0.19 0.22 0 0.04 0.01
90 0.08 0.13 0.14 0.11 0.04 0.02 0 0.00 0.00
99 0.00 0.02 0.04 0.02 0.01 0.01 0 0.00 0.00

MC gains percentiles: Chance of gains from first to last period.
_a is simulation from estimated distribution of returns of mix.
_b is mix of simulations from estimated distribution of returns from individual funds.

Velliv_m Velliv_m_long Velliv_h PFA_m PFA_h mix_m_a mix_h_a mix_m_b mix_h_b
0 94.84 96.82 96.14 97.93 99.21 99.03 100.00 99.67 99.85
5 94.24 96.25 95.77 97.63 99.10 98.90 100.00 99.60 99.84
10 93.54 95.78 95.37 97.45 99.04 98.78 100.00 99.53 99.80
25 90.97 94.02 93.79 96.57 98.62 97.98 100.00 99.19 99.70
50 85.58 89.93 91.12 94.54 97.60 96.26 99.98 97.71 99.40
100 71.77 78.01 83.82 87.56 94.75 89.49 99.69 90.46 97.44
200 39.90 44.79 64.07 58.34 85.78 59.77 93.12 48.74 86.31
300 15.77 17.51 44.93 21.85 71.26 22.11 71.97 11.33 65.34
400 4.79 4.56 29.03 3.92 55.59 3.58 44.50 1.22 40.59
500 1.28 1.07 17.26 0.39 39.63 0.20 23.12 0.11 21.15
1000 0.00 0.00 0.70 0.00 2.50 0.00 0.28 0.01 0.09

Summary statistics

Fit summary

Summary for fit of log returns to an F-S skew standardized Student-t distribution.
m is the location parameter.
s is the scale parameter.
nu is the estimated degrees of freedom, or shape parameter.
xi is the estimated skewness parameter.

Velliv_medium Velliv_medium_long Velliv_high PFA_medium PFA_high mix_medium mix_high
m 0.048 0.052 0.065 0.058 0.084 0.059 0.082
s 0.120 0.115 0.150 0.123 0.121 0.088 0.071
nu 3.304 2.706 3.144 2.265 3.185 2.773 89.863
xi 0.034 0.505 0.002 0.477 0.018 0.029 0.770
R-squared 0.993 0.978 0.991 0.991 0.964 0.890 0.961

Monte Carlo simulations summary

Monte Carlo simulations of portfolio index values (currency values).
Statistics are given for the final state of all paths.
Probability of down-and_out is calculated as the share of paths that reach 0 at some point. All subsequent values for a path are set to 0, if the path reaches at any point.
0 is defined as any value below a threshold.
losing_prob_pct is the probability of losing money. This is calculated as the share of paths finishing below index 100.

## Number of paths: 10000
Velliv_m Velliv_m_long Velliv_h PFA_m PFA_h mix_m_a mix_m_b mix_h_a mix_h_b
mc_m 278.437 292.991 405.817 319.768 555.977 324.319 301.885 501.245 475.563
mc_s 122.559 116.159 217.397 104.054 244.760 97.388 82.893 158.271 163.327
mc_min 1.432 0.000 0.000 0.000 0.626 0.000 18.971 129.151 25.692
mc_max 960.495 1003.923 1483.003 999.664 1748.121 673.260 1997.325 1335.386 1204.922
dao_prob_pct 0.000 0.010 0.010 0.010 0.000 0.010 0.000 0.000 0.000
losing_prob_pct 5.160 3.180 3.860 2.070 0.790 0.970 0.330 0.000 0.150

Ranking:

mc_m ranking mc_s ranking mc_min ranking mc_max ranking dao_prob_pct ranking losing_prob_pct ranking
555.977 PFA_h 82.893 mix_m_b 129.151 mix_h_a 1997.325 mix_m_b 0.00 Velliv_m 0.00 mix_h_a
501.245 mix_h_a 97.388 mix_m_a 25.692 mix_h_b 1748.121 PFA_h 0.00 PFA_h 0.15 mix_h_b
475.563 mix_h_b 104.054 PFA_m 18.971 mix_m_b 1483.003 Velliv_h 0.00 mix_m_b 0.33 mix_m_b
405.817 Velliv_h 116.159 Velliv_m_long 1.432 Velliv_m 1335.386 mix_h_a 0.00 mix_h_a 0.79 PFA_h
324.319 mix_m_a 122.559 Velliv_m 0.626 PFA_h 1204.922 mix_h_b 0.00 mix_h_b 0.97 mix_m_a
319.768 PFA_m 158.271 mix_h_a 0.000 Velliv_m_long 1003.923 Velliv_m_long 0.01 Velliv_m_long 2.07 PFA_m
301.885 mix_m_b 163.327 mix_h_b 0.000 Velliv_h 999.664 PFA_m 0.01 Velliv_h 3.18 Velliv_m_long
292.991 Velliv_m_long 217.397 Velliv_h 0.000 PFA_m 960.495 Velliv_m 0.01 PFA_m 3.86 Velliv_h
278.437 Velliv_m 244.760 PFA_h 0.000 mix_m_a 673.260 mix_m_a 0.01 mix_m_a 5.16 Velliv_m

Appendix

Average of returns vs returns of average

Math

\[\text{Avg. of returns} := \dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2}\] \[\text{Returns of avg.} := \left(\dfrac{ x_t + y_t }{2}\right) \Big/ \left(\dfrac{ x_{t-1} + y_{t-1} }{2}\right) \equiv \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

For which \(x_1\) and \(y_1\) are \(\text{Avg. of returns} = \text{Returns of avg.}\)?

\[\dfrac{ \left(\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} \right) }{2} = \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

\[\dfrac{x_t}{x_{t-1}} + \dfrac{y_t}{y_{t-1}} = 2 \dfrac{ x_t + y_t }{ x_{t-1} + y_{t-1}}\]

c{ x_t + y_t }{ x_{t-1} + y_{t-1}} \[ \](x_{t-1} + y_{t-1}) x_t y_{t-1} + (x_{t-1} + y_{t-1}) x_{t-1} y_t = 2 (x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)$$

\[(x_{t-1}x_1y_{t-1} + y_{t-1}x_ty_{t-1}) + (x_{t-1}x_{t-1}y_t + x_{t-1}y_{t-1}y_t) = 2(x_{t-1}y_{t-1}x_t + x_{t-1}y_{t-1}y_t)\] This is not generally true, but true if for instance \(x_{t-1} = y_{t-1}\).

Example

Definition: R = 1+r

## Let x_0 be 100.
## Let y_0 be 200.
## So the initial value of the pf is 300 .
## Let R_x be 0.5.
## Let R_y be 1.5.

Then,

## x_1 is R_x * x_0 = 50.
## y_1 is R_y * y_0 = 300.

Average of returns:

## 0.5 * (R_x + R_y) = 1

So here the value of the pf at t=1 should be unchanged from t=0:

## (x_0 + y_0) * 0.5 * (R_x + R_y) = 300

But this is clearly not the case:

## 0.5 * (x_1 + y_1) = 0.5 * (R_x * x_0 + R_y * y_0) = 175

Therefore we should take returns of average, not average of returns!

Let’s take the average of log returns instead:

## 0.5 * (log(R_x) + log(R_y)) = -0.143841

We now get:

## (x_0 + y_0) * exp(0.5 * (log(Rx) + log(Ry))) = 259.8076

So taking the average of log returns doesn’t work either.

Simulation of mix vs mix of simulations

Test if a simulation of a mix (average) of two returns series has the same distribution as a mix of two simulated returns series.

## m(data_x): 0.027182 
## s(data_x): 0.4186434 
## m(data_y): 9.355286 
## s(data_y): 3.453421 
## 
## m(data_x + data_y): 4.691234 
## s(data_x + data_y): 1.725588

m and s of final state of all paths.
_a is mix of simulated returns.
_b is simulated mixed returns.

m_a m_b s_a s_b
94.184 93.787 7.893 7.847
93.971 94.041 7.776 7.904
93.715 93.716 7.879 8.076
93.682 94.041 7.704 7.661
93.840 93.319 7.775 7.979
93.618 94.168 8.049 7.707
93.877 93.763 7.777 7.614
93.573 93.890 7.850 7.732
93.656 94.022 7.724 7.806
93.901 93.801 7.736 7.505 
##       m_a             m_b             s_a             s_b       
##  Min.   :93.57   Min.   :93.32   Min.   :7.704   Min.   :7.505  
##  1st Qu.:93.66   1st Qu.:93.77   1st Qu.:7.746   1st Qu.:7.672  
##  Median :93.78   Median :93.85   Median :7.777   Median :7.769  
##  Mean   :93.80   Mean   :93.85   Mean   :7.816   Mean   :7.783  
##  3rd Qu.:93.90   3rd Qu.:94.04   3rd Qu.:7.872   3rd Qu.:7.890  
##  Max.   :94.18   Max.   :94.17   Max.   :8.049   Max.   :8.076

_a and _b are very close to equal.
We attribute the differences to differences in estimating the distributions in version a and b.

The final state is independent of the order of the preceding steps:

So does the order of the steps in the two processes matter, when mixing simulated returns?

The order of steps in the individual paths do not matter, because the mix of simulated paths is a sum of a sum, so the order of terms doesn’t affect the sum. If there is variation it is because the sets preceding steps are not the same. For instance, the steps between step 1 and 60 in the plot above are not the same for the two lines.

Recall, \[\text{Var}(aX+bY) = a^2 \text{Var}(X) + b^2 \text{Var}(Y) + 2ab \text{Cov}(a, b)\]

var(0.5 * vhr + 0.5 * phr)
## [1] 0.005355618
0.5^2 * var(vhr) + 0.5^2 * var(phr) + 2 * 0.5 * 0.5 * cov(vhr, phr)
## [1] 0.005355618

Our distribution estimate is based on 13 observations. Is that enough for a robust estimate? What if we suddenly hit a year like 2008? How would that affect our estimate?
Let’s try to include the Velliv data from 2007-2010.
We do this by sampling 13 observations from vmrl.

##        m                 s          
##  Min.   :0.05940   Min.   :0.04151  
##  1st Qu.:0.06477   1st Qu.:0.06170  
##  Median :0.06704   Median :0.06573  
##  Mean   :0.06897   Mean   :0.06776  
##  3rd Qu.:0.07281   3rd Qu.:0.07030  
##  Max.   :0.08353   Max.   :0.09079